Parliamentary Efficiency under Majority and Supermajority Rules: The Role of Independent Legislators

Abstract

Parliaments dominated by two political blocs often face legislative inefficiencies as polarization increases. A central institutional question concerns how majority and supermajority rules interact with parliamentary composition to balance governability and minority protection. This article examines how the inclusion of independent legislators, introduced through sortition, affects collective decision-making under different majority and supermajority quorum requirements. Using an agent-based model that combines analytical threshold derivations with numerical simulations, we identify four critical thresholds that partition the parameter space into distinct coalition regimes. These regimes range from majority-party dominance to minority veto and, at high levels of diversity, to structural fragmentation in which coordination becomes increasingly difficult. Parliamentary efficiency emerges from non-linear interactions among quorum thresholds, party-size distribution, and the proportion of independents. Under simple majority, the system exhibits an interior efficiency maximum associated with a transition from unilateral control to pivot-based coalition formation. Under strong supermajorities, however, the same increase in diversity may induce minority veto dynamics or coordination breakdown, significantly reducing legislative performance. These results show that institutional performance is not an intrinsic property of a particular decision rule but an emergent outcome of the interaction between approval thresholds, parliamentary composition, and the coalition structures they generate.

Figures (15)

• Figure 1: Distribution of agents and acceptance windows in the Cartesian plane. Example distribution of independent legislators ($\bigstar$) and political party affiliates ($\blacktriangle$ and $\blacktriangledown$) within the circle defined by the center $C$ ($\bullet$) and the tolerance radius $r$. The square area enclosed by the dotted line corresponds to the possible locations of the party centers. The shaded areas represent the acceptance windows for the majority party and an independent legislator. The legislative proposal $a_n$ ($\times$) lies in the line $y_n$, and their x position depends on the agent that votes for them. In this figure, all legislators and parties below the line $y_n$ approve the proposal $a_n$ because their acceptance window contains the proposal's position.
• Figure 2: Voting rules are based on the affiliation of the legislator submitting the proposal. Each independent legislator considers voting in favor of the proposal if it is within his or her window of acceptance, in addition to voting in favor of the proposals he or she makes. On the other hand, a party's legislator will vote in favor of a proposal if the proposal is within the party's acceptance window or if the proponent belongs to the same party.
• Figure 3: Schematic visualization of the dynamics of the model. A legislature starts with a random distribution of independent agents and agents belonging to parties. Then, a legislative act begins, selecting a random agent and proposing a law that will be voted on by all the agents in the system, following the voting rules. After verifying whether the law is approved or rejected, another legislative act is carried out until there is a complete legislature. Finally, the metrics used to evaluate the parliament's performance are calculated.
• Figure 4: Parliamentary dynamics under simple majority in a polarized scenario ($p=0.55$, $q=1/2$). a) Phase diagram showing the five phases ($I$–$V$) determined by the analytical thresholds $A$, $B$, $C$, and $D$. The horizontal dashed line indicates the value of $p=0.55$ used in this scenario. Phases correspond to distinct coalition structures required to reach the quorum. b) Average parliamentary efficiency $\langle E\!f\!f \rangle$ as a function of the number of independent legislators $N_{ind}$. Vertical dashed lines mark the threshold values separating phases. c) Representative composition of the parliament in each phase, showing the proportion of majority party, minority party, and independent legislators. The dashed horizontal line indicates the quorum required for legislative proposal approval ($50\%+1$). Simulation parameters: $N=1000$, $N_a=1000$, $r=0.2$, and $1000$ legislative terms averaged for each data point. The figure shows that increasing the number of independent legislators leads to successive regime shifts in the coalition structure needed to reach the approval quorum. Parliamentary efficiency peaks between Phases $II$ and $III$, where independents improve the social welfare of approved proposals without yet causing coordination breakdown.
• Figure 5: Parliamentary dynamics under strong supermajority in a polarized scenario ($p=0.55$, $q=2/3$). a) Phase diagram illustrating the phases intersected by the horizontal line corresponding to $p=0.55$. In contrast to the simple majority case, the system operates only in Phases $III$–$V$ due to the higher approval threshold. Phases correspond to distinct coalition structures required to reach the quorum. b) Average parliamentary efficiency $\langle Eff \rangle$ as a function of the number of independent legislators $N_{ind}$. Vertical dashed lines denote the analytical thresholds separating phases. c) Representative parliamentary composition in each phase, displaying the proportions of majority party, minority party, and independent legislators. The dashed horizontal line indicates the required two-thirds quorum for legislative proposal approval. Simulation parameters: $N=1000$, $N_a=1000$, $r=0.2$, and $1000$ legislative terms averaged per configuration. The figure shows that under a two-thirds quorum, the system structurally needs broad multi-party coalitions, since no single party can reach the approval quorum on its own. Parliamentary efficiency remains relatively stable while a two-party structure persists (Phases $III$–$IV$), but collapses once independents dominate the chamber (Phase $V$), where forming a supermajority becomes increasingly unlikely.